Calculate Per-User Usage Charges

Problem Statement

You are building a billing engine that calculates a user's monthly invoice based on a list of events and a pricing_plan. Each event has a timestamp (integer), type, and quantity.

The billing period is defined from start_time to end_time (exclusive).

Rules:

1. Base Fee: Every user pays a flat monthly fee of base_price.
2. Usage Tiers: Usage is billed according to tiers:
   • Tier 1: 0 to 100 units = $0.50/unit
   • Tier 2: 101+ units = $0.25/unit
3. Proration: If a user joins mid-month, the base_price is prorated based on the fraction of the month they were active (total active time / total month duration). Usage-based charges are not prorated.

Function Signature

def calculate_invoice(events, pricing_plan, start_time, end_time, user_join_time):
    # events: list of dicts {'timestamp': int, 'quantity': int}
    # pricing_plan: dict {'base_price': float}
    # returns: float representing the final bill

Constraints

• Time is represented as Unix integers.
• events are not guaranteed to be sorted.
• user_join_time is within the [start_time, end_time] range.

Example

Input: events = [{'timestamp': 100, 'quantity': 60}, {'timestamp': 200, 'quantity': 60}] pricing_plan = {'base_price': 100.0} start_time = 0, end_time = 400, user_join_time = 200

Calculation:

• Proration: Active time = (400-200) = 200. Total month = 400. Factor = 200/400 = 0.5. Prorated Base = 100 * 0.5 = 50.0.
• Usage: Total units = 120. First 100 at 0.5 ($50), remaining 20 at 0.25 ($5) = $55.0.
• Total: 50.0 + 55.0 = 105.0.

Edge Cases

• No events recorded during the period.
• User joins exactly at end_time.
• Usage exactly at the tier boundary (100 units).